1. Introduction


title

body

body

body

body


caption –

Terms Defined

Calculating the Mean Squared Error
for a SKATER Model


Euclidean dissimilarity, \(d\):
For two polygons, \(p_i\) and \(p_j\), the Euclidean distance in high-dimensional attribute space (for variables \(1\) to \(n\)) is: \[\begin{equation} d(p_{i},p_{j}) = \sqrt{(p_{i1} - p_{j1})^2 + (p_{i2} - p_{j2})^2 + ... + (p_{in} - p_{jn})^2} \end{equation}\]

Mean intraregion dissimilarity, \(D\):
For a region \(R\) with \(m\) polygons (\(p\)), add up all the pairwise Euclidean dissimilarities for the region and divide by the number of polygons in the region: \[\begin{equation} D(R) = \frac{\sum\limits_{i=1}^{m-1}\sum\limits_{j=i+1}^m d(p_{i},p_{j})}{m} \end{equation}\]

Error, \(E\):
For a region, \(R\), with \(m\) polygons (\(p\)), the error, \(E\), is found by taking the sum of each pairwise Euclidean distance (\(d_{ij}\)) minus the mean intraregion dissimilarity (\(D\)): \[\begin{equation} E(R) = \sum\limits_{i=1}^{m-1}\sum\limits_{j=i+1}^m \Big(d(p_{i},p_{j}) - D(R)\Big) \end{equation}\]

Mean squared error, \(MSE\):
For a SKATER model with \(K\) regions (\(R\)), the mean squared error, \(MSE\), is the average squared error for all the regions in the model: \[\begin{equation} MSE(K) = \frac{\sum\limits_{k=1}^K E(R_k)^2}{K} \end{equation}\]

---
title: "MathJax 2"
author: "Erin M. Ochoa"
date: "04/08/2019"
output: 
  flexdashboard::flex_dashboard:
    storyboard: true
    theme: flatly
    source_code: embed
  mathjax: https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-MML-AM_CHTML
---

```{r setup}
library(sf)
library(sp)
library(spdep)
library(rgdal)
#library(leaflet)
library(mapview)
#library(htmltools)
library(tidyverse)
#library(RColorBrewer)
```

```{r style_mathjax}
# Use MathJax; tweak the nav bar; define styles for text slides
# https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/latest.js?config=TeX-MML-AM_CHTML
```




### 1. Introduction {data-commentary-width=450}

```{r frame1}
```

***

title

body

body

body

body


caption –

### Terms Defined {data-commentary-width=0}

Calculating the Mean Squared Error
for a SKATER Model


Euclidean dissimilarity, $d$:
For two polygons, $p_i$ and $p_j$, the Euclidean distance in high-dimensional attribute space (for variables $1$ to $n$) is: $$\begin{equation} d(p_{i},p_{j}) = \sqrt{(p_{i1} - p_{j1})^2 + (p_{i2} - p_{j2})^2 + ... + (p_{in} - p_{jn})^2} \end{equation}$$

Mean intraregion dissimilarity, $D$:
For a region $R$ with $m$ polygons ($p$), add up all the pairwise Euclidean dissimilarities for the region and divide by the number of polygons in the region: $$\begin{equation} D(R) = \frac{\sum\limits_{i=1}^{m-1}\sum\limits_{j=i+1}^m d(p_{i},p_{j})}{m} \end{equation}$$

Error, $E$:
For a region, $R$, with $m$ polygons ($p$), the error, $E$, is found by taking the sum of each pairwise Euclidean distance ($d_{ij}$) minus the mean intraregion dissimilarity ($D$): $$\begin{equation} E(R) = \sum\limits_{i=1}^{m-1}\sum\limits_{j=i+1}^m \Big(d(p_{i},p_{j}) - D(R)\Big) \end{equation}$$

Mean squared error, $MSE$:
For a SKATER model with $K$ regions ($R$), the mean squared error, $MSE$, is the average squared error for all the regions in the model: $$\begin{equation} MSE(K) = \frac{\sum\limits_{k=1}^K E(R_k)^2}{K} \end{equation}$$